science.sciencemag.org/content/366/6465/637/suppl/DC1

Supplementary Materials for

A noninteracting low-mass black hole–giant star binary system

Todd A. Thompson*, Christopher S. Kochanek, Krzysztof Z. Stanek, Carles Badenes,

Richard S. Post, Tharindu Jayasinghe, David W. Latham, Allyson Bieryla, Gilbert A.

Esquerdo, Perry Berlind, Michael L. Calkins, Jamie Tayar, Lennart Lindegren, Jennifer

A. Johnson, Thomas W.-S. Holoien, Katie Auchettl, Kevin Covey

*Corresponding author. Email: thompson.1847@osu.edu

Published 1 November 2019, Science 366, 637 (2019)

DOI: 10.1126/science.aau4005

This PDF file includes:

Materials and Methods

Figs. S1 to S9

Tables S1 to S8

References

1 Materials and Methods

1.1 Object selection method

The APOGEE survey, which is part of SDSS-IV (34, 35), provides multi-epoch spectroscopy

for over ⇠ 105 stars in the Galaxy (13, 17, 36). A catalog of high signal-to-noise radial velocity

(RV) measurements was assembled by (14). In general, there are ⇠ 2 � 4 measurements per

system. Although a measured difference in RV between subsequent epochs can indicate the

presence of a binary companion, the orbit is in general not well-established with such a small

number of RV samplings (37, 38). A simple criterion is used to identify systems that might

have a massive unseen companion. We first calculated the maximum acceleration amax for each

system,

amax =�RV

�tRV

����max

(S1)

where �RV is the difference between the measured RV in subsequent epochs and �tRV is

the time between the two observations. Because many systems exhibit no RV differences larger

than the uncertainties in the APOGEE RV determinations, we limited our exploration to systems

with �RV > 1 km s�1 (14). The measured maximum acceleration then gives an estimate for

the unseen companion mass M(amax) ⇠ amaxs2/G, where s is the separation between the

two bodies over the time �tRV. The separation s is unknown, but in the absence of other

information we used s = �RV�tRV for the two epochs that determine amax, which yields

an expression similar to the binary mass function M(amax) = �RV3�tRV/G. This does not

accurately reproduce the unseen companion mass, but is instead a very simple first method to

prioritize 105 systems.

We then acquired existing photomertic observations from the public ASAS-SN database

(15, 16) for the ⇠ 200 systems with the highest M(amax) as estimated from the APOGEE data.

Our goal was to estimate the orbital period for some systems using the photometric variability

2

expected from ellipsoidal variations, eclipses, or starspots. Many systems showed no variability,

while others showed periodic photometric variations.

The system in our sample with the longest well-measured photometric period was the gi-

ant star 2MASS (Two Micron All Sky Survey (39)) J05215658+4359220, with coordinates

RA(J2000) = 05 21 56.591 and Dec(J2000) (deg) = +43 59 21.958 (23). The raw aperture

photometry lightcurve from the ASAS-SN Sky Patrol lightcurve server (29) is shown in Figure

S1. The phased lightcurve is shown in Figure 1.

We use the Generalized Lomb-Scargle (GLS) periodogram method (40, 41) to derive the

photometric period (Pphot) of the ASAS-SN light curve, finding a best-fitting period of Pphot '

82.2 day. To estimate the uncertainty in the period, we calculated the full width at half maximum

(FWHM) of the GLS periodogram peak to be ' 4.9 day. Using the Multi-Harmonic Analysis

Of Variance periodogram (42), we derive a photometric period consistent with that obtained

from the GLS periodogram. Although the best-fitting photometric period differs from the best-

fitting RV period (P = 83.205± 0.064 day; Table S2) by ⇠ 1%, the results for the two periods

are consistent within the uncertainties.

Given the 3 APOGEE measurements for J05215658 listed in Table S2, we estimated the RV

semi-amplitude K ⇠ (42.6+37.4)/2 ⇠ 40 km s�1. Assuming that the orbital period is equal to

the photometric period Porb = Pphot (starspots in a tidally locked binary) or that Porb = 2Pphot

(as expected for ellipsoidal variations) we estimated the mass function f(M) > 0.6M� (eq.

1). Assuming that the observed giant had a mass larger than 1 M�, the implied minimum

companion mass is above the Chandrasekhar mass of ' 1.4M�. Given the absence of any

evidence for a stellar companion (see below), we initiated RV followup to measure the orbital

period and multi-band photometric followup to determine both a densely-sampled lightcurve

and to constrain potential color variations.

3

Figure S1: Un-phased ASAS-SN lightcurve. Raw aperture photometry ASAS-SN lightcurveof J05215658 from the public Sky Patrol lightcurve server (16), showing the V -band flux as afunction of Heliocentric Julian Day (HJD). The photometric periodicity is evident. Comparewith Figure 2A.

4

1.2 Multi-Band Optical Photometric Followup

To establish the nature of the photometric variability of J05215658, we obtained multi-band

(BVri) images at the Post Observatory Mayhill (New Mexico, USA), which employs a robotic

0.61m CDK (Corrected Dall-Kirkham) telescope with a back illuminated Apogee U47 cam-

era. Between 12 September 2017 and 14 February 2018 (about 155 days), between 90 and

120 60-sec images were obtained in each band. To extract an instrumental light curve in each

filter, we used standard image subtraction procedures (43, 44). As our target star is a red giant,

the resulting light curves have photon-noise precision that is better than 1% in Vri-filters and

approximately 1% in the B-band filter. The resulting lightcurve is shown in Figure 2A. The

average calibrated magnitudes derived from these observations are given in Table S8.

1.3 Gaia Parallax, Binary Motion, and Proper Motion1.3.1 Parallax, Offset, and Uncertainty

The parallax listed in Data Release 2 (DR2) (23) from the Gaia satellite for J05215658 (source

ID 207628584632757632) is 0.272 ± 0.049 mas, implying a distance of 3.7 kpc. However,

it is known (23) that the parallaxes in Gaia DR2 have a negative zero point, which should

be subtracted from the catalog values, thus making the parallaxes larger. This zero point is

approximately �0.030mas for faint quasars, and varies by about 0.043mas RMS (root-mean

square) depending on the position of the object on the sky (23). There is also evidence that the

zero point varies with magnitude and that it is more negative for bright Gaia G-band sources

with G < 15. For example, Ref. (45) find an offset of �0.046± 0.013mas for bright Cepheids

(G ⇠ 9), Ref. (46, 47) find an offset of �0.0528 ± 0.0024 ± 0.001mas from a comparison

with asteroseismic data for stars in the Kepler field with G ⇠ 12, Ref. (48) find an offset of

�0.083 ± 0.033mas from bright eclipsing binaries (G . 12), and Ref. (49) find an offset of

�0.057mas for RR Lyrae stars with G ⇠ 12. The variation in the zero point offset for bright

5

sources as a function of position on the sky is not known, but we assume that it is similar to the

faint variation: 0.043mas RMS.

Given the Gaia DR2 catalogue value of 0.272± 0.049mas, and assuming a zero point offset

of �0.050 given the Gaia magnitude G ' 12.3 for J05215658, we take the measured parallax

to be

⇡ = 0.322mas± 0.049mas (random)± 0.043mas (systematic), (S2)

implying a nominal distance of D ' 3.1+0.8�0.5 kpc. For comparison, the analysis of (50) gives a

distance of D ' 3.3+0.6�0.5 kpc.

1.3.2 Binary Motion and Parallax Bias

We considered the possibility that the parallax given in equation (S2) could be biased by the

orbital motion of the system. The semi-major axis of the orbit of the giant around the center

of mass is s = a[MCO/(MCO + Mgiant)], where a is the semi-major axis of the relative orbit.

Using the observed mass function f(M) and the period P ,

s =

✓Gf(M)P 2

4⇡2 sin3 i

◆1/3

' 0.34AU

sin i

✓f(M)

0.77M�

◆1/3 ✓P

83.2 days

◆2/3

, (S3)

so that the ratio of the angular orbital motion to the parallax is

orbital motion

parallax' 0.34

sin i

✓f(M)

0.77M�

◆1/3 ✓P

83.2 days

◆2/3

. (S4)

For a nominal parallax of ⇡ ' 0.322mas (eq. S2), equation (S4) then implies that the binary

motion will be 0.34⇥ 0.322mas ' (0.11mas)/ sin i.

In order to estimate what effect this motion might have on the measured parallax, we made

a Monte Carlo simulation of a circular orbit with period 83.2 days. The orientation of the orbit

in space was taken to be random, while the phase of the motion was constrained according

to the RV curve (Figure 2). We then used the timing and geometric scan angle data from the

6

Gaia Observation Forecast Tool (GOFT) file (51) for J05215658 to calculate the parallax biases

resulting from the binary motion by least-squares fitting of the five astrometric parameters to

the calculated orbital displacement of each transit across the focal plane. The 26 GOFT entries

cataloging the number of transits span more than 500 days and a wide range of scan angles,

but half of the observations occurred in a 5 day timespan with two sets of observations in that

period highly clustered in time and scan angle.

Figure S2 shows the results for 104 model systems, randomly sampling the unknown angles

associated with the system’s projection on the sky. In panel (A) the fractional parallax bias

of the single-star astrometric solution caused by the orbital motion of the binary is plotted

against sin i for the 104 systems. The bias scales with the size of the orbit and hence with

the parallax according to equation (S4), and is therefore expressed as a fraction of the true

parallax. Our sign convention for the parallax bias is such that a negative bias means that the

true parallax is larger than the measured parallax, whereas a positive parallax bias means that

the true parallax is smaller than the measured parallax. We find that there could be a positive

or negative parallax bias, and that the fractional size of the effect can be substantial ⇠ ±1 for

nearly face-on configurations with sin i ⇠ 0.

Figure S2B shows the fractional bias converted into units of mas by multiplying by an

assumed true parallax of ⇡ = 0.322mas (eq. S2) and histograms of the parallax bias for sin i >

0.0, > 0.6, and > 0.8. The bias has strong peaks at ±0.026mas. For sin i > 0.8, the bias

is essentially never larger than ±0.07mas, whereas for nearly face-on configurations a small

percentage of systems could in principle have large biases. However, as we show below, such

large parallax biases are ruled out by the measured goodness of fit in the single-star astrometric

solution from Gaia, even without a constraint on sin i.

Cases with positive (negative) parallax bias will make the measured parallax larger (smaller)

than the true one, and therefore correspond to the case where the actual distance, stellar radius,

7

and luminosity are greater (smaller) than implied by equation (S2). Because the parallax bias

in the left panel of Figure S2 scales with the true parallax, we ask the following question: how

large (small) can the true parallax reasonably be, such that the bias from the orbital motion,

combined with the observational uncertainties in equation (S2), could lead to a measured value

as small (large) as 0.322mas? Our answer is listed in Table S1, which gives the true parallax

values at which a given fraction of the simulated cases results in a measured parallax that is

or � 0.322mas. Including all sin i > 0, the 1-� (68% confidence interval) parallax is

⇡ ' 0.322+0.086�0.074, (S5)

which is the value used in the main text. To be precise, this means that if the true parallax is

in fact 0.322 + 0.086 = 0.408 or 0.322 � 0.074 = 0.248, then the probability of obtaining

an observed value 0.322 or � 0.322, respectively, is (1 � 0.68)/2 = 0.16 (1-�). For the

2-� parallax (95% confidence interval), we find that ⇡ ' 0.322+0.224�0.146 for all sin i > 0. It is

not possible to set an upper 3-� limit on the true parallax for sin i > 0, because the fractional

parallax bias is . �1 in more than 0.13% of the cases (see Figure S2); these systems could

therefore in principle have an arbitrarily large true parallax and still be consistent with the

measured small value. As we show below, these very large values of the parallax bias are ruled

out by the astrometric goodness of fit.

Restricting to more nearly edge-on configurations with sin i > 0.8, our determination of

the confidence intervals for the parallax are ⇡ ' 0.322+0.075�0.069 (1-�), ⇡ ' 0.322+0.155

�0.134 (2-�), and

⇡ ' 0.322+0.242�0.199 (3-�). These values are subject to our assumption of a zero-point offset of

�0.050mas relative to the reported Gaia DR2 parallax of 0.272mas.

1.3.3 Constraints from the Astrometric Goodness of Fit

Our simulations show that a large parallax bias is accompanied by a large increase in the RMS

residual to the single-star astrometric solution, which could in principle appear in DR2 as

8

Table S1: Parallax bias constraints. Confidence intervals for the parallax relative to ⇡ =0.322mas, including contributions from statistical and systematic uncertainties (eq. S2), andthe binary orbital motion (Figs. S2, S3). For a normal distribution, the fractions in the secondcolumn correspond to the number of standard deviations in the first column.

deviation fraction ⇡ (mas) ⇡ (mas) ⇡ (mas) ⇡ (mas)sin i > 0.0 sin i > 0.2 sin i > 0.6 sin i > 0.8

�3 �. 0.9787 0.095 0.117 0.122 0.123�2 � 0.9772 0.176 0.181 0.186 0.188�1 � 0.8413 0.248 0.249 0.252 0.253+1 � 0.1587 0.408 0.406 0.399 0.397+2 � 0.0228 0.546 0.527 0.486 0.477+3 � 0.0013 – 1.470 0.588 0.564

astrometric excess noise or an increased astrometric chi-square. J05215658 has zero excess

noise, which nominally means that the single-star model fits the data well; however, the ex-

cess noise is problematic in DR2 and a better goodness-of-fit indicator is the Renormalized

Unit Weight Error (RUWE) (52). The RUWE is essentially the square root of the reduced chi-

square, and should be around 1.0 for good single-star astrometric solutions. J05215658 has

RUWE = 1.055, which indicates that additional noise from orbital motion is small.

Because the binary motion has to be relatively small in comparison with the random errors

per observation in order to get a RUWE close to unity, the observed RUWE can be used to rule

out systems with large contributions to the RMS residual from binary motion. From the GOFT

data described earlier we find that the stated DR2 parallax uncertainty corresponds to an RMS

uncertainty per observation of 0.28mas. The astrometric residuals caused by the orbital motion

must be considerably smaller than this, or about 0.1mas, to be consistent with the observed

RUWE. Because the residuals caused by the orbital motion scale with the size of the orbit, this

in turn sets a limit on the angular size of the orbit and hence on the parallax.

9

To quantify this argument we made additional simulations where, for each random system,

the expected RUWE was computed in addition to the parallax bias. A sample of results are

shown in Figure S3. Each point shows the computed RUWE and measured parallax for a

random simulated system with true parallax equal to 0.3mas (Fig. S3A), 0.5mas (Fig. S3B),

0.7mas (Fig. S3C), and 0.9mas (Fig. S3D). Consistent with our expectations, we see that a

large offset between the measured parallax and the true parallax (the parallax bias) comes with

an increase in the value of the RUWE.

In Figure S3, each panel is constructed from a sample of 104 modeled systems at the given

true parallax. Figure S4 summarizes the results from 100 times more simulations to improve

the statistics. As a function of the true parallax, Figure S4A shows the fraction of systems

that have measured parallax 0.322mas for different limits on sin i. Figure S4B shows the

fraction of systems with measured parallax 0.322mas and RUWE 1.1, i.e. conservatively

consistent with the observed goodness of fit of RUWE = 1.055. Even without a constraint on

sin i the small observed value of the RUWE for J05215658 rules out a true parallax greater than

' 0.62mas at the 3-� confidence level; for sin i > 0.6 or > 0.8, on the other hand, the RUWE

does not substantially affect the limits in Table S1.

1.3.4 Proper Motion

Gaia measures proper motions in the right ascension (RA) direction of pmra = �0.055 ±

0.10mas yr�1 and in the declination direction of pmdec = �3.69 ± 0.07mas yr�1, imply-

ing a total proper motion µ ' 3.69 ± 0.07mas yr�1 and a total tangential velocity of '

52.5 km s�1 (D/3 kpc), substantially larger than the absolute radial velocity of 3.56 ± 0.1

km s�1 from the TRES analysis (Section 1.4). Taking into account the correlation coeffi-

cient pmra pmdec corr = �0.490, in Galactic coordinates, these proper motions translate

to µl = 3.01 ± 0.10mas yr�1 in the Galactic longitude direction and µb = �2.13 ± 0.07mas

10

yr�1 in the Galactic latitude direction. The simulations discussed above using the GOFT data

to assess the parallax bias show that the proper motion components are not likely to be biased

by more than a few tenths of a mas yr�1 from the binary orbital motion.

The J-band reduced proper motion (RPM) for J05215658 is RPMJ = J+log(µ) ' �2.33,

where J ' 9.83 is the 2MASS magnitude (Table S8) and µ is the total proper motion in arc-

seconds per year. With a color of J � H ' 0.76, the system falls among the giants in a re-

duced proper motion diagram (53). We also used the Gaia Universe Model Snapshot (GUMS)

simulation (54) to compare the observed proper motion of J05215658 with the expected mo-

tions of stars at various distances from the Sun. Extracting sources in GUMS within 5 degrees

of J05215658, and with similar apparent G magnitude (±1mag) and color index GBP � GRP

(±0.25mag), we find that the observed proper motion is typical for a giant at a distance of

1.5� 5 kpc from the Sun, but improbably small for a more nearby dwarf.

1.4 Radial Velocity Followup

We initiated spectroscopy with TRES (55) on the 1.5m Tillinghast Reflector at the Fred Lawrence

Whipple Observatory (FLWO) located on Mt. Hopkins in Arizona. TRES has a spectral reso-

lution of R ⇠ 44, 000 and spectra were collected using the medium 2.3” fiber.

A total of 11 spectra were obtained between 10 September 2017 and 25 January 2018.

The spectra were reduced and extracted as described in (56). The exposure times ranged from

30� 42 minutes depending on observing conditions and yielded an average signal-to-noise per

resolution element (SNRe) of ⇠ 25 at the peak of the continuum centered at 519 nm surround-

ing the Mg b triplet. We derived relative radial velocities using the observation with the highest

SNRe as a template and cross-correlated the remaining spectra order-by-order against the ob-

served template. The multi-order velocity analysis avoids including orders that have substantial

contamination by telluric lines. The uncertainties for the multi-order velocities are based on the

11

Figure S2: Parallax bias of the single-star astrometric solution as a result of the binarymotion for J05215658. (A) Fractional parallax bias as a function of sin i for 104 simulatedsystems (see text and Table S1). (B) Histograms of the number of simulated systems per binversus the parallax bias in mas, assuming a true parallax of ⇡ = 0.322mas (eq. S2), for sin i >0.0 (black, solid) > 0.6 (red, dashed), and > 0.8 (blue, dotted). The parallax bias is typically' ±0.026mas, and less than ±0.07mas for sin i > 0.8.

12

¡0:5 0 0:5 1:0 1:5

Measured parallax [mas]

0:6

0:8

1:0

1:2

1:4

1:6

1:8

2:0

RU

WE

¡0:5 0 0:5 1:0 1:5

Measured parallax [mas]

0:6

0:8

1:0

1:2

1:4

1:6

1:8

2:0

RU

WE

¡0:5 0 0:5 1:0 1:5

Measured parallax [mas]

0:6

0:8

1:0

1:2

1:4

1:6

1:8

2:0

RU

WE

¡0:5 0 0:5 1:0 1:5

Measured parallax [mas]

0:6

0:8

1:0

1:2

1:4

1:6

1:8

2:0

RU

WE

A B

C D

Figure S3: RUWE for model systems of J05215658, including binary motion. The computedRUWE is shown as a function of the measured parallax for an assumed true parallax of 0.3 mas(A), 0.5 mas (B), 0.7 mas (C), and 0.9 (D). Solid lines mark the measured RUWE= 1.055 andmeasured parallax of 0.322mas (eq. S5). Because a large absolute value of the parallax biasis accompanied by a large increase in the RMS residual to the single-star astrometric solution(Fig. S2), the low value of the measured RUWE rules out the possibility that the true parallaxcould be & 0.62mas (see Figure S4).

13

0:3 0:4 0:5 0:6 0:7 0:8 0:9 1:0

True parallax (mas)

0:001

0:002

0:005

0:01

0:02

0:05

0:1

0:2

0:5

1

Fra

ctio

n

sin i > 0:0sin i > 0:2sin i > 0:4sin i > 0:6sin i > 0:8

0:3 0:4 0:5 0:6 0:7 0:8 0:9 1:0

True parallax (mas)

0:001

0:002

0:005

0:01

0:02

0:05

0:1

0:2

0:5

1

Fra

ctio

n

sin i > 0:0sin i > 0:4sin i > 0:6sin i > 0:8

Figure S4: Parallax consistency with measured values. Fraction of model systems for whichthe computed parallax and astrometric goodness of fit are consistent with observed values, asa function of the assumed true parallax of the system. Panel (A) shows the fraction of systemswith measured parallax 0.322mas, whereas Panel (B) shows the fraction of systems withmeasured parallax 0.322mas and RUWE 1.1. The curves are for a given range of sin ifrom > 0.0 (red; thickest) to > 0.8 (magenta; thinnest). The horizontal dashed lines show thefractions corresponding to the upper 1-, 2-, and 3-� confidence limits in Table S1. The nearlyface-on cases (small sin i) in Panel (A) consistent with a large true parallax are ruled out by theobserved RUWE in Panel (B).

14

velocity scatter, order by order. The observed template is, by definition, assigned a velocity of

0 km s�1.

The absolute velocity of the system is 3.56 ± 0.1 km s�1. This is derived from the radial

velocity for the template observation when correlated against our library of calculated spectra

using the Mg b order, combined with a �0.61 km s�1 correction, which is mostly due to the fact

that the calculated template spectrum does not include gravitational redshift. The uncertainty is

based on residual systematics from observations of the International Astronomical Union (IAU)

Radial Velocity Standard Stars. The derived relative radial velocity results from TRES are given

in Table S2 and Figure 2B. Note that the absolute APOGEE radial velocities listed in Table S2

were not included in the fit reported in Table S3. We experimented with including the APOGEE

data in the RV fit using the code radvel (57), and found results consistent with those reported

in Table S3, but since the APOGEE data likely have larger systematic uncertainties than the

⇠ 10m s�1 reported in Table S2 (14), and since there are likely additional systematic offsets

between the two instruments, we prefer to report the derived relative radial velocity results from

TRES alone.

1.5 Properties of the Giant

Archival and new photometry of the system is summarized in Table S8. In addition to the

data we collected as part of our multi-band photometric followup, we obtained Swift Ultravio-

let/Optical Telescope (UVOT) and X-ray Telescope (XRT) imaging, which yielded a detection

in the U and UVM2 bands and an upper limit in the X-ray. The UVM2 detection is impor-

tant for constraining a stellar companion, as discussed in Section 1.6. The X-ray upper limit is

discussed in Section 1.7.

15

Table S2: Radial Velocity Measurements from APOGEE and TRES.

APOGEE:

HJD Absolute RV Uncertainty�2450000 (km s�1) (km s�1)

6204.95544 �37.417 0.0116229.92499 34.846 0.0106233.87715 42.567 0.010

TRES:

HJD Relative RV Uncertainty�2450000 (km s�1) (km s�1)

8006.97517 0.000 0.0758023.98151 �43.313 0.0758039.89955 �27.963 0.0458051.98423 10.928 0.1188070.99556 43.782 0.0758099.80651 �30.033 0.0548106.91698 �42.872 0.1358112.81800 �44.863 0.0888123.79627 �25.810 0.1158136.59960 15.691 0.1468143.78352 34.281 0.087

1.5.1 Analysis of TRES Spectra

We used the Stellar Parameter Classification (SPC) tool to derive stellar parameters from the

TRES spectra (58) discussed above. SPC cross correlates the observed spectra against a library

of synthetic spectra based on Kurucz model atmospheres (59). We ran SPC on each TRES

spectrum individually and then report the results as a weighted average. The weighted average

16

Table S3: Orbital Parameters derived from TRES RV measurements listed in Table S2.

Parameter Fitted Value Units

Orbital period P 83.205± 0.064 daysTime of primary center T 8115.93± 7.4 HJD�2450000Orbital eccentricity e 0.00476± 0.00255Argument of periastron ! 197.13± 32.07 degreesRadial velocity semi-amplitude K 44.615± 0.123 km s�1

TRES radial velocity offset � �0.389± 0.101 km s�1

Binary mass function f(M) 0.766± 0.00637 M�

results from this analysis are Te↵ = 4574 ± 65K, log g = 2.35 ± 0.14, and a metallicity

of [m/H] = �0.39 ± 0.08, where [m/H] denotes the logarithmic metal abundance relative

to the Solar value (58). The total line broadening parameter is Vbroad = 16.8 ± 0.6 km s�1,

which accounts for the instrumental broadening of TRES (6.8 km s�1), but does not distinguish

between the contributions from rotational broadening and macroturbulence.

Ref. (60) (see also Ref. (61)) suggests an empirical fit to results from high-resolution spec-

troscopic data to connect the total line broadening parameter and its contributions from rotation

and macroturbulence for giants of the form

Vbroad =⇥(v sin i)2 + 0.95⇠2RT

⇤1/2, (S6)

where ⇠RT is the unknown radial-tangential macroturbulent dispersion. Using this relation, in

Table S4 we report the implied value of v sin i for values of ⇠RT ranging from 0�10 km s�1. Ref.

(60) also provides an empirical relation between the total broadening parameter and a higher

resolution determination of v sin i (their equation 2) for their sample of both red giant branch

and red horizontal branch stars. Directly applying their relation with the TRES determination

of Vbroad, we obtain v sin i ' 13.4 km s�1. Given the quoted scatter of 1.5 km s�1 in this

17

relation, this determination of v sin i is in agreement with the APOGEE v sin i (see below) and

with the range of ⇠RT considered in Table S6. No red giant in the sample of Ref. (60) has

⇠RT greater than ' 10 km s�1, and those that have ⇠RT ' 10 km s�1 are more luminous than

J05215658 for the nominal Gaia parallax (see Section 1.3). The set of giants with properties

closest to J05215658 in Ref. (60) have ⇠RT ⇠ 4 � 7 km s�1. Indeed, the empirical fits to ⇠RT

summarized in equations 4, 5, and 6, of Ref. (60) give estimates for ⇠RT of 5.1, 5.5, and 5.6 km

s�1 for J05215658. These numbers should be compared with the range used for ⇠RT in Table

S4. Assuming a tidally circularized and synchronized system, as in the main text, we use the

TRES measurements to derive the minimum giant radius and luminosity, distance and parallax,

and mass from log g ' 2.35 ± 0.14 (M log ggiant = g R2/G). Values of all parameters are reported

in the Table. Similar estimates are made for the APOGEE v sin i below and in Table S5.

For ⇠RT = 0� 8 km s�1, Table S4 shows that the implied parallax from v sin i is ⇠ 0.343�

0.387, consistent with ⇡ = 0.322+0.086�0.074 from Gaia (eq. S5; Table S1). For the largest values of

⇠RT considered in Table S4, the minimum mass inferred from log g is & 3.0M�, implying a

companion of MCO & 3.0M� (Figure 3). The value of ⇠RT required for M log ggiant to equal 1M�,

as implied by the near-Solar [C/N] abundance (see main text and Section 1.5.3), is ⇠RT '

15.8 km s�1 (for log g = 2.35).

1.5.2 Analysis of APOGEE Spectra

Analysis of J05215658 using the APOGEE Stellar Parameter and Chemical Abundances Pipeline

(ASPCAP) (62) yields stellar parameters of Te↵ ' 4480.0 ± 62.3K, log g ' 2.59 ± 0.06,

[M/H] ' �0.298 ± 0.03, [↵/M] ' �0.04 ± 0.015, and [C/N] ' 0.0. The Cannon analy-

sis (63) of the spectra gives similar results: Te↵ ' 4406.4 ± 57K, log g ' 2.653 ± 0.136,

[M/H] ' �0.309 ± 0.059, [↵/H] ' �0.052 ± 0.043, where the latter two measurements of

the abundance measure logarithm of the total metallicity and total ↵-element abundance with

18

Table S4: Effects of macroturbulent broadening on the giant using TRES. Properties of thegiant for assumed values of ⇠RT = 0�10 km s�1 (eq. S6), and assuming that the binary is tidallysynchronized, and combined with Te↵ , log g, and Vbroad derived from the TRES spectroscopy.The quantity M log g

giant = g R2/G.

⇠RT v sin i R sin i L sin2 i D sin i parallax / sin i M log ggiant sin

2 i(km s�1) (km s�1) (R�) (L�) (kpc) (mas) (M�)

0.00 16.8 27.6 301 2.92 0.343 6.23 +2.37�1.72

2.00 16.7 27.5 297 2.90 0.345 6.15 +2.34�1.69

4.00 16.3 26.9 285 2.84 0.352 5.90 +2.24�1.62

6.00 15.7 25.9 264 2.74 0.365 5.48 +2.08�1.51

8.00 14.9 24.5 236 2.59 0.387 4.89 +1.86�1.35

10.0 13.7 22.5 200 2.38 0.421 4.13 +1.57�1.14

respect to Solar values (62).

We employed the analysis technique used by Ref (25) to determine the projected rotational

velocity of the giant in J05215658 from the APOGEE spectra. Figure S5 shows a piece of

the APOGEE spectrum, as well as model spectra including macroturbulence, broadened with

v sin i = 0.0, 5.0, and 14.1 km s�1 as well as the residuals. Using this method over the full

APOGEE spectral range, we find v sin i = 14.1± 0.6 km s�1. This estimate for v sin i includes

macroturbulence through the fitting function proposed by (64), which gives a macroturbulent

broadening parameter of ' 3.6 km s�1. For stars like J05215658, the distribution of macrotur-

bulent broadening parameters is tightly clustered to a sequence near 4 km s�1, but with some

outliers (64) (their figure 9).

As in the discussion of the TRES spectroscopy (see above), we considered the possibility

that the macroturbulence was underestimated for J05215658. Given the uncertainties in the

macroturbulent broadening parameter, we view the TRES and APOGEE v sin i determinations

as reasonably consistent (i.e., TRES: [16.82� 0.95⇠2RT]1/2 versus APOGEE: 14.1 km s�1). Still,

19

Table S5: Effects of macroturbulent broadening on the giant using APOGEE. Properties ofthe giant using Vbroad derived from the APOGEE spectroscopy, Te↵ from the SED (Figure S8),and assuming different values of ⇠RT (eq. S6), and that the binary is tidally synchronized. Thequantity M log g

giant = g R2/G uses the TRES value of log g since the APOGEE log g measurementis likely biased (see text).

⇠RT v sin i R sin i L sin2 i D sin i parallax / sin i M log ggiant sin

2 i(km s�1) (km s�1) (R�) (L�) (kpc) (mas) (M�)

0.00 14.5 23.9 216 2.47 0.405 4.64 +1.77�1.28

2.00 14.4 23.6 212 2.45 0.408 4.56 +1.73�1.26

4.00 14.0 23.0 200 2.38 0.420 4.31 +1.64�1.19

6.00 13.3 21.8 181 2.26 0.442 3.89 +1.48�1.07

8.00 12.2 20.1 153 2.08 0.480 3.30 +1.26�.909

10.0 10.7 17.7 118 1.83 0.547 2.54 +.968�.701

for completeness, and as a guide to how an underestimate in ⇠RT might change the results,

we used equation (S6) to estimate the value of the total broadening parameter if the assumed

3.6 km s�1 had not been subtracted, yielding Vbroad ' 14.5 km s�1. Then, as in the discussion of

macroturbulent broadening with TRES above, we used equation (S6) to estimate the change in

our derived minimum stellar radius and luminosity using values of ⇠RT ranging from 0� 10 km

s�1 (60). Table S5 shows the results for the parameters derived. The mass determined from log g

is computed using the TRES log g ' 2.35±0.14 since the APOGEE log g is likely substantially

biased (see below; Fig. S6). We find that for ⇠RT as large as 10 km s�1, the implied giant mass

is larger than 1.8M�, implying a minimum mass for the companion of MCO & 2.4M� (Fig. 3).

The values of v sin i and ⇠RT also enter our comparison to single-star evolutionary models

in Section 1.5.4 because once a best-fitting stellar radius is determined by comparing to the

models, a value of sin i can be inferred to derive the range of acceptable companion masses.

APOGEE log g: The large value of log g ' 2.59 ± 0.06 found by APOGEE ASPCAP

20

Figure S5: Rotational velocity. Example of the change in the fit to a piece of the observedAPOGEE spectral lines (black thick solid) as the projected rotation velocity of the model spec-trum is increased from v sin i = 0.0 (solid thin gray line), 5.0 (dotted blue), and 14.1 km s�1

(dashed red). The bottom set of lines shows the residuals with respect to the data. Similarimprovements to the model fit are indicated throughout the APOGEE spectrum. We find a bestfit of v sin i = 14.1 ± 0.6 km s�1. Corrections for macroturbulent broadening are discussed inSection 1.5.2 and Table S5.

21

Data Release 14 (DR14) (62) is substantially different than the value obtained from the optical

TRES spectra, log g ' 2.35 ± 0.14. To address this question we compared spectroscopic log g

determinations from APOGEE with those determined from asteroseismology from the Apache

Point Observatory-Kepler Asteroseismology Science Consortium (APOKASC) sample (65).

We were particularly interested in whether or not rapidly rotating giants might have systemati-

cally discrepant spectroscopic log g as a function of v sin i (64). Figure S6 shows the difference

between the spectroscopic log g and the asteroseismic log g from the APOKASC sample for

systems with well-measured v sin i (25) as a function of v sin i, spectroscopic log g (APOGEE),

spectroscopic Te↵ , and iron abundance [Fe/H]. The data points are represented with circles

whose size is proportional to v sin i for clarity. We find that the large majority of the data are

above 0, indicating that the spectroscopic log g determination is systematically larger than the

asteroseismic determination. There is additionally a trend with v sin i such that the difference

between the spectroscopic and asteroseismic log g determinations increases with v sin i. For

v sin i ' 8� 14 km s�1, the offset is 0.1� 0.5 dex. Because of the systematic trend in this com-

parison and the large potential systematic offset, we opt to use the TRES log g determination in

the main text even though it does not employ a correction for macroturbulent broadening (see

Section 1.4; Table S4). A bias in the APOGEE log g value could affect other spectroscopically

determined parameters for J05215658 discussed in Section 1.5.3.

Other Analyses: The analysis by Ref. (66) finds a higher value for the effective temperature

of Te↵ ' 4645.7K, similar [Fe/H] ' �0.311 with [↵/Fe] ' 0.159, and a lower value of

log g ' 2.220 than APOGEE ASPCAP. The low value of log g may have been inferred from

the fact that J05215658 has a lower Te↵ than the majority of giants at the nominal APOGEE

log g (see Section 1.5.4) and a near-Solar value of [C/N] (Section 1.5.3). Indeed, the analysis

of Ref (66) yields a very low mass of ln(M/M�) ' �0.6671 (' 0.51M�) and an unphysical

age of ln(Age/Gyr) ' 4.331 (76 Gyr). These values are inconsistent with our determinations

22

of the giant radius and luminosity from the Gaia distance, the argument from v sin i and tidal

synchronization, and the measured proper motions (see Section 1.3).

Ref. (67) use Gaia and APOGEE, among other spectroscopic surveys, and a Bayesian frame-

work to characterize the probability density of distance, mass, and age for giants throughout

the Galaxy. For J05215658, they find a distance of 1.465 kpc (parallax 0.683) and a mass of

1.24M�. Their quoted distance is inconsistent with our 2-� upper limits on the parallax for any

value of sin i in Table S1. As we show in Section 1.3, and in Figures S3 and S4, the low noise

of the Gaia single-star astrometric solution essentially rules out a true parallax as large as 0.683.

1.5.3 Abundances

The derived abundances from APOGEE Data Release 13 (DR13) (64,68,69) are shown in Table

S6. The system is observed to be metal poor, but with near-Solar [C/N] ' 0.034, and modest

enhancements in S and O with respect to Fe of [S/Fe] ' 0.244 and [O/Fe] ' 0.118.

Figure S9 shows the [C/N] abundances as a function of asteroseismic mass from the APOKASC

sample (65). Giant stars as massive as we infer for J05215658 are rare in APOGEE, with just

135 stars having asteroseismic masses > 2.5M� in the 6700 APOKASC database. As discussed

in the main text, the fact that J05215658 has [C/N] ' 0.0 implies a low Mgiant ' 1M� in the

absence of other information. This lower value for the giant mass is highlighted in Figure 3 by

the region labelled “[C/N].” The bias in the APOGEE log g discussed in Section 1.5.2 (Fig. S6)

may affect the abundance determinations and other spectroscopic parameters.

Three stars in the APOKASC sample have asteroseismic Mgiant > 2.0M� and [C/N] �

�0.1 (KIC 8649099, KIC 11954055, and KIC 9541892 with asteroseismic masses of 2.0, 2.7,

and 3.1 M�, respectively). Although such objects are well away from the mean trend in [C/N]

versus mass, a fraction of the massive stars in the sample also have high [C/N]. Such stars may

be the result of stellar mergers (70). The fraction of stars in the sample with high values of

23

Figure S6: Bias in log g for rapidly rotating giant stars in APOGEE. All panels show thedifference between the APOGEE DR14 (64) log g and the log g as measured by asteroseis-mology as a function of v sin i from the APOKASC sample (65), matched to those stars withv sin i measurements from (25). In each panel, the size of the circle is scaled by the the valueof v sin i. Panels (A), (B), (C), and (D) show this difference in log g as a function of v sin i,APOGEE log g, Te↵ , and [Fe/H], respectively. These plots demonstrate that APOGEE system-atically overestimates log g for rapidly rotating stars in the range of parameters appropriate forJ05215658 (v sin i ' 14 km s�1, Te↵ ' 4500K, [Fe/H] ' �0.4, log g (APOGEE)' 2.6). Panel(A) shows that there is a systematic trend as a function of v sin i.

24

Table S6: APOGEE DR13 Abundances for J05215658 (69).

Element Abundance Uncertainty

[Al/H] �0.769 0.069[Ca/H] �0.457 0.034[C/H] �0.321 0.041[Fe/H] �0.503 0.036[K/H] �0.629 0.059[Mg/H] �0.428 0.028[Mn/H] �0.355 0.041[Na/H] �0.518 0.102[Ni/H] �0.402 0.042[N/H] �0.355 0.074[O/H] �0.385 0.027[Si/H] �0.605 0.040[S/H] �0.259 0.042[Ti/H] �0.633 0.048[V/H] �0.632 0.135

[C/N] apparently increases as a function of the giant mass: while only 8/1577 ' 0.005 of all

> 1.5M� giants in APOKASC have [C/N] > �0.1 the fraction increases to 2/135 ⇠ 0.015 for

Mgiant > 2.5M� and to 1/18 ⇠ 0.06 for Mgiant > 3.0M�. The shaded regions in Figure S9

denote the 1- and 2-� confidence intervals of Mgiant from Figure 3.

1.5.4 Comparison with Single-Star Theoretical Evolutionary Tracks

Figure S7 shows L and log g versus Te↵ for stellar evolutionary models of different masses

and for [Fe/H] = �0.4 and 0.0. The shaded regions indicate Te↵ ' 4525.0 ± 90K, log g =

2.35±0.14, and L = 331+227�125 L�, as inferred from TRES, the Gaia parallax, and the bolometric

flux and SED. The low value of the effective temperature may be interpreted as favoring a lower

mass giant of ⇠ 1M�, whereas the bolometric luminosity strongly favors a higher mass giant

25

of ⇠ 2� 3M� when comparing by-eye to the MIST tracks.

Quantitative fits to single-star evolutionary tracks are discussed in the main text. Given the

nominal values and uncertainties in L, Te↵ , and R from the Gaia parallax and our fit to the

SED, and using the TRES value of log g ' 2.35 ± 0.14 as a constraint, we find a best fitting

value of Mgiant ' 3.2+1.0�1.0 M� (2-�). Using the fitted value of the giant radius R, and comparing

with the minimum radius obtained from v sin i, we derive a constraint on sin i that allows us to

constrain the companion mass to be MCO ' 3.3+2.8�0.7 M�. For a given assumed value of v sin i,

some fraction of the fitted evolutionary models have an unphysical sin i > 1. As discussed

in the main text, if we assume a value of the macroturbulent broadening large enough to give

a v sin i = 10 km s�1, corresponding to ⇠RT > 10 km s�1 in Table S5, more lower-mass giant

models have physical values of sin i < 1, and best-fit giant masses of ' 1.8M� can be obtained.

However, these lower-mass giant models do not have much lower companion masses because

the lower implied value of sin i drives up the companion mass MCO, in accord with the mass

function (Fig. 3). For the models we have explored, this leads to MCO > 2.5M�.

Our results do not change qualitatively if we change the constraint on log g, or impose no

constraint on log g at all. In both cases, we find that the best fits for the giant mass decrease to

the lower end of the range quoted when imposing the TRES log g, with best-fit giant masses in

the range of Mgiant ' 2.2� 2.5M�. Imposing v sin i, these fits then give sin i ' 0.8� 0.9 and

best-fitting values of the compact object companion mass MCO ' 2.9� 4.0M�.

1.5.5 Limits on the Giant Radius from Ellipsoidal Variations

We do not convincingly detect ellipsoidal modulations in the ASAS-SN lightcurve. Using a

periodogram search and Lomb-Scargle analysis, the ASAS-SN lightcurve exhibits a small peak

in power at a period of ' 83.2/2 days as expected for ellipsoidal variations, but when we

subtract the dominant periodicity associated with the spot modulation, we find a phase for the

26

modulation that is inconsistent with ellipsoidal variations: the maximum photometric variation

is different from the maximum RV blueshift by ' 30 degrees. To assess whether we could detect

ellipsoidal variations of a given amplitude, we injected periodic modulations into the ASAS-SN

photometry (Figs. 1 and S1) consistent with the phase and period of the radial velocity curve for

ellipsoidal variations of specified V -band amplitude, using the cadence and photometric errors

from the actual ASAS-SN observations. Then, using the same types of searches, we look for

power with the specified period and phase. Using these explorations, we find that the signal at

' 83.2/2 days in the ASAS-SN lightcurve has peak-to-peak amplitude of order ' 3%, and that

this is close to the minimum we could detect.

With this upper bound on the photometric variations for which we would expect to see evi-

dence of periodic modulations consistent with ellipsoidal variability, we can construct a bound

on the giant radius (71)

R3 =3.070A3 �M (3� u)

q sin2 i (⌧ + 1)(15 + u), (S7)

where A is the semi-major axis in Solar radii, q = MCO/Mgiant, �M is the peak-to-peak

variation in the lightcurve, u ' 0.83 is the limb-darkening coefficient, ⌧ ' 0.46 is the gravity-

darkening coefficient assuming a late-type star with a convection envelope, Te↵ = 4500K, and

V -band observations. For �M . 0.03, we find that R . 30R� for sin i = 1 and q = 1.

In our fitting to the evolutionary tracks described in the main text, we find that the gi-

ant radii derived over our preferred parameter regime are always small enough that they can

accommodate the peak-to-peak variability limit of ' 3% we infer from the ASAS-SN photom-

etry. However, these fits in general produce stellar radii, component masses, and semi-major

axes predicting that the ellipsoidal variability should appear at the ⇠ 1% level.

27

1.5.6 Spectral Energy Distribution & UV Detection

Figure S8 shows the SED of the system. We fit the Wide-field Infrared Survey Explorer (WISE)

3.4 and 4.6µm (72), 2MASS J, H and Ks (39), our BVri and the U -band Swift photome-

try using model atmospheres with metallicity [Z/H] = �0.5 and log g = 2.5 (59), assuming

10% flux uncertainties to compensate for variability, the Gaia distance of 3.11 kpc (parallax

0.322 mas; eq. S5), an extinction law with RV = 3.1 (73) (where RV = AV /(AB � AV ) and

AV and AB are the extinctions in the V and B photometric bands, respectively), and a spec-

troscopic temperature of Te↵ = 4550± 100K (Section 1.5). This process yielded a luminosity

of log(L/L�) = 2.52 ± 0.03, a temperature of Te↵ = 4530 ± 89K, and interstellar reddening

E(B�V ) = AB�AV ' 0.445±0.050. The photometry slightly improves the constraint on the

temperature, and the reddening is consistent with estimates for this distance from three dimen-

sional dust maps (74). Figure S8 shows the spectral energy distribution of the giant and our best

fitting model at the nominal Gaia distance. The goodness of fit is �2 = 5.83 for 8 degrees of

freedom. This fit to the SED at the spectroscopic temperature and including substantial Galactic

reddening with the standard RV = 3.1 extinction law also implies that the dust properties in the

direction of J05215658 are not unusual. The WISE 12 and 22µm fluxes lie on the red extension

of the SED model so there is no infrared excess indicating the presence of circumstellar dust

and extinction. The bolometric flux used throughout the text is F ' 1.1⇥ 10�9 ergs cm�2 s�1.

Figure S8 shows the best-fitting model SED at the Gaia distance of 3.11 kpc and SEDs for

main sequence companions of 1.3, 1.4, 1.5, and 1.8 M� at an age of 1 Gyr for comparison. Also

shown is the sum of the main sequence models and the best-fitting SED to demonstrate how a

main sequence companion would effect the bluer bands at the Gaia distance. If the parallax bias

induced by the binary orbital motion is negative (see Section 1.3), and the true parallax of the

system is larger, then the fitted luminosity of the giant decreases and the spectral distortions to

the bluer bands caused by assuming a main sequence companion increase (see Section 1.6).

28

UV Detection: The Swift UVM2 observation was not included in the SED fit. The mean

magnitude from our observations is reported in Table S8 and shown for the Gaia distance in Fig-

ure S8, but we were unable to find a satisfactory SED fit that included it. The Galaxy Evolution

Explorer (GALEX) satellite reports a near-ultra violet (NUV) detection at ' 21.5 ± 0.4mag,

which we also include in Figure S8, falling approximately a factor of ⇠ 3 below the mean Swift

UVM2 flux. In addition, we find evidence for variability in the UVM2 band from the multi-

epoch Swift follow-up observations. Table S7 gives the date of the Swift observation, derived

magnitude or upper limit, and uncertainty in each observation. While our first observation gave

a > 20.27mag 3-� upper limit, subsequent observations yield detections of ' 20.2� 19.8mag.

The brightness variation is inconsistent with the known multi-band variability from Figure 2A in

both amplitude and phase. We calculated the Swift UVM2 fluxes for two other stars in the field

with sky locations RA(J2000) = 05 21 58.466 and Dec(J2000) = +43 50 54.113 (Star 1) and

RA(J2000) = 05 22 04.92 and Dec(J2000) = +43 58 54.801 (Star 2) during our observations

to check if the UVM2 variability of J05215658 might be an artifact of the observations or image

processing. Star 2 showed virtually constant flux over all observations, varying by 0.12mag,

while Star 1 showed variability at the level of 0.51mag, but opposite the implied trends derived

for J05215658: while Star 1 became brighter from one epoch to the next, J05215658 became

dimmer.

We thus conclude that J05215658 is variable in the UVM2 Swift band. A number of inter-

pretations can be considered. In Section 1.6 we consider the possibility of a stellar companion,

and show that main sequence or stripped envelope stars cannot simultaneously explain the UV

photometry and the RV curve. Wind-fed accretion onto a neutron star or black hole could also

be considered, but the X-ray upper limit described in Section 1.7 constrains this possibility.

The simplest interpretation of the variable UVM2 detection is that J05215658 has some level of

stellar activity, which is common for rapidly rotating giants. Giants with Te↵ and log g similar

29

Table S7: Multi-epoch Swift UVM2 Photometry.

Observation Date UVM2 3-� uncertainty(HJD-2450000) (Vega mag) (mag)

8083.552 > 20.278339.848 19.75 0.218341.311 20.22 0.218343.498 19.90 0.198345.564 20.06 0.228369.536 19.77 0.18

to J05215658 commonly exhibit both UV excesses and variability (75).

1.6 Limits on a Stellar Companion

As shown in Figure 3, for sin i = 1 and Mgiant = 1M�, the minimum mass of the unseen

companion allowed from the radial velocity measurements is ' 1.8M�. Figure S8 shows the

spectral energy distributions of 1.3, 1.4, 1.5, and 1.8M� main sequence stars at 1 Gyr (59)

compared to our fit to the photometry at the nominal Gaia distance of 3.11 kpc.

The Swift UVM2 detection puts constraints on possible companions (Table S8), ruling out

main sequence stellar companions of > 1.4M�. While a lower mass < 1.4M� companion

is nominally consistent with the UVM2 limit, this mass is inconsistent with the results from

Figure 3 unless the giant mass is Mgiant ' 0.2M� (0.5M�) for sin i = 0.9 (sin i = 1.0). Such

a low value for the giant mass is implausible given the distance to the system, the luminosity of

the giant, and its evolutionary state.

As discussed in Section 1.3 there may be a bias in the Gaia parallax as a result of astrometric

binary orbital motion (see eq. S5). If the distance is in fact smaller than the nominal value of

3.1 kpc used for Figure S8, then the photometric limits on a main sequence companion become

30

tighter because the data points and SED model would move to lower luminosity, while the main

sequence stellar models would remain unchanged. For example, if the distance to the system

was D ' 2.0 kpc instead of 3.1 kpc, the 1.4M� main sequence companion would be excluded

and the 1.3M� model would be at the mean Swift UVM2 detection. However, the U , B, and

V bands would then be a poor fit, and it would be impossible to explain the radial velocity

variation. Indeed, we can only accommodate a higher-mass companion photometrically if the

distance is underestimated by Gaia. For example, for a 1.8M� main sequence companion star,

which would satisfy the dynamical constraints from the RV measurements if Mgiant = 1M�

(sin i = 1), to be consistent with the photometry, the distance to the system would need to be

> 2 times larger than 3.1 kpc, and the luminosity of the giant would then need to be more than

4 times larger, over 103 L�. Such a luminosity would then be inconsistent with a giant mass

as low as the Mgiant = 1M� needed to accommodate a companion of 1.8M�. We therefore

see no way to have a main sequence companion that satisfies the dynamical, photometric, and

astrometric constraints on the system. As discussed in Section 1.5.6, the UVM2 detection, and

its variability, is more reasonably interpreted as intrinsic variability of the rapidly rotating giant

star.

Much cooler stellar companions that evade the Swift UVM2 detection are inconsistent with

the SED unless they have the same temperature as the giant. In that case, the companion would

also have to be a giant star, and we would then expect it to produce absorption lines in the TRES

and APOGEE spectroscopy. However, neither the TRES nor the APOGEE spectra (Section 1.4)

show any evidence for a second set of spectral lines at any of the radial velocity epochs (Figure

2; Table S2). Moreover, in all of the allowed parameter space of Figure 3 for Mgiant < 3M�, the

unseen putative red giant companion would have to be more massive than the observed giant.

Finally we see no evidence of an excess in the NIR SED that might be evidence of a massive

cooler companion.

31

One can also consider much hotter companions. We considered whether stripped stellar

cores might be able to meet the dynamical and photometric constraints. For example, in Ref.

(76) a 1.8M� stripped core has a bolometric luminosity of ⇠ 103 � 103.5 L� and an effective

temperature of 50�60 kK. Such a model would exceed the Swift UVM2 detection in Figure S8

by a factor of ⇠ 100, dominate the U -band flux, and contribute to the bluer optical bands. At

fixed luminosity of ⇠ 103 � 103.5 L�, the effective temperature of such a stripped core would

have to be > 4 � 5 times higher to accommodate the Swift UVM2 detection. In addition, for

the high luminosities expected for a 1.8M� core, bright optical emission lines associated with

the strong ionizing flux would be expected to arise, but which are not present in the observed

spectra. We conclude that it is not possible to satisfy both the dynamical and photometric

constraints with a stripped very hot stellar core.

1.7 X-Ray Upper Limit & Wind-Fed Accretion

For an interacting black hole or neutron star binary can produce X-ray emission from accretion.

While there are weak limits on the X-ray emission from J05215658 from the Roentgensatel-

lit (ROSAT) All-Sky Survey (RASS) (77), we obtained much stronger limits from the Swift

X-ray Telescope (XRT; (78–80)) observations made simultaneously with the first UVOT obser-

vations discussed in Section 1.6. This observation (ObsID:00010442001), taken on 2017-11-26

01:07:21 Coordinated Universal Time (UTC), was reprocessed from level one XRT data using

the Swift XRTPIPELINE version 0.13.2 script (81), and with the most up to date calibration files,

following standard filter and screening criteria suggested by the Swift collaboration (82).

We find no evidence for X-ray emission associated with J05215658 to the upper limit re-

ported in Table S8. There is a faint nearby X-ray point source located at RA(J2000) =

05 21 56.6 and Dec(J2000) = +43 49 22, approximately 30 arcsec away from the system.

To minimize contamination from this nearby source when deriving our 3-� count-rate upper

32

limit, we use a source region with a radius of 20 arcsec centered on J05215658 and a source

free-background region centered at RA(J2000) = 05 21 30.9 and Dec(J2000) = +43 56 40.3

with a radius of 150 arcsec. Correcting for the fraction of the total counts from the system

that would be enclosed by our source region (a 20 arcsec radius corresponds to an encircled

energy fraction of ⇠ 80% at 1.5 keV assuming on-axis pointing (83)), we obtain an upper limit

of 5 ⇥ 10�4 counts s�1 in the 0.3 � 10.0 keV energy band. Assuming an absorbed powerlaw

with a photon index of � = 2, and a Galactic HI column density of 4.03 ⇥ 1021 cm�2 derived

from (84), this count rate corresponds to an unabsorbed flux limit of 4.4⇥ 10�14 ergs s�1 cm�2,

or ' 1 ⇥ 10�2 L� at 3.1 kpc, roughly 107 times smaller than the Eddington luminosity for a

3 M� black hole.

We considered the possibility of wind-fed accretion from the giant to the compact ob-

ject companion. We scale the wind mass loss rate to Mwind,�10 = Mwind/10�10 M� yr�1,

with a wind velocity of Vwind, 200 = Vwind/200 km s�1, approximately the escape velocity for

a 3M� star with R = 25R�. The total separation between the two bodies we take to be

s = 0.68 au/ sin i (eq. S4). An estimate of the amount of material gathered at the sphere of

influence of the compact object is

Macc ⇠Mwind

(4⇡s2)⇡

✓GMCO

V 2wind

◆2

⇠ 2⇥ 10�13 M� yr�1Mwind,�10 M

2CO, 3 sin

2 i

V 4wind, 200

(S8)

where MCO, 3 = MCO/3M�. For radiatively efficient accretion onto a black hole, we would

expect an accretion luminosity of order ⇠ 0.1Maccc2 ⇠ 0.35L�. Although this is above the

X-ray upper limit, for such low accretion rates the flow will be radiatively inefficient (85). For

a massive neutron star companion with even a small surface dipole magnetic field, the energy

density of the field greatly exceeds ram pressure of the accreted material at the neutron star

surface, implying that much of the gas may be expelled from the system without accreting.

33

Figure S7: Comparison to single-star evolutionary tracks. Bolometric luminosity and log gas a function of Te↵ for MIST single-star models with [Fe/H] = �0.4 and [Fe/H] = 0 fora range of masses from 1 � 5M� (86–89). Gray bands indicate properties of the giant withTe↵ = 4525 ± 90K and log g = 2.35 ± 0.14 (Section 1.5.1). The horizontal dashed line andgray band indicate the bolometric luminosity L ' 331+227

�125 L� of the giant as inferred in themain text from the Gaia distance and observed bolometric flux. See discussion in main text andSection 1.5.2.

34

0.1 1 10

1

10

100

Figure S8: Broadband SED of J05215658. Observed SED normalized for the nominal Gaiadistance of 3.11 kpc (eq. S2; data points). The best fitting model is shown as the red solid line,as described in Section 1.5.6, with fit parameters labelled. The blue dashed lines show SEDs formain sequence companions of 1.3, 1.4, 1.5, and 1.8 M� for comparison. The dotted black linesshow the sum of the best-fitting SED and the main sequence models. Reddened giant templatescannot accommodate the UVM2 Swift detection. See Sections 1.5.6 and 1.6.

35

Figure S9: The Carbon-to-Nitrogen abundance ratio as a function of mass. The [C/N]ratio from APOGEE is shown as a function of the asteroseismic mass in the APOKASC sample(65). The [C/N] abundance of J05215658 is measured by APOGEE to be ' 0.0 (Table S6).The shaded region and the black empty circle show the mass ranges and best fitting value forMgiant from Figure 3. The bias in the APOGEE log g determination (see Section 1.5.2, Fig. S6)may affect the abundance determinations for J05215658 and other rapidly rotating giants in theAPOGEE sample.

36

Table S8: New and Archival Photometry

Instrument Band Magnitude Uncertainty Referenceor Facility or Filter or Flux (cgs)

WISE F34W 8.73 0.05 (72)F46W 8.79 0.05 (72)

2MASS Ks 8.88 0.05 (39)H 9.07 0.05 (39)J 9.83 0.05 (39)

Post Observatory i 11.64 0.05 this workr 12.27 0.05 this workV 12.89 0.05 this workB 14.34 0.05 this work

Swift UVOT U (Vega) 15.56 0.04 this workUVM2 (Vega) 20.00 0.095 this work

Swift XRT 0.3� 10 keV < 4.4⇥ 10�14 this work

Further Acknowledgement

SDSS-IV acknowledges support and resources from the Center for High-Performance Com-

puting at the University of Utah. The SDSS web site is www.sdss.org. SDSS-IV is managed

by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Col-

laboration including the Brazilian Participation Group, the Carnegie Institution for Science,

Carnegie Mellon University, the Chilean Participation Group, the French Participation Group,

Harvard-Smithsonian Center for Astrophysics, Instituto de Astrofısica de Canarias, The Johns

Hopkins University, Kavli Institute for the Physics and Mathematics of the Universe (IPMU) /

37

University of Tokyo, Lawrence Berkeley National Laboratory, Leibniz Institut fur Astrophysik

Potsdam (AIP), Max-Planck-Institut fur Astronomie (MPIA Heidelberg), Max-Planck-Institut

fur Astrophysik (MPA Garching), Max-Planck-Institut fur Extraterrestrische Physik (MPE),

National Astronomical Observatories of China, New Mexico State University, New York Uni-

versity, University of Notre Dame, Observatario Nacional / MCTI, The Ohio State University,

Pennsylvania State University, Shanghai Astronomical Observatory, United Kingdom Partici-

pation Group, Universidad Nacional Autonoma de Mexico, University of Arizona, University

of Colorado Boulder, University of Oxford, University of Portsmouth, University of Utah, Uni-

versity of Virginia, University of Washington, University of Wisconsin, Vanderbilt University,

and Yale University.

This work has made use of data from the European Space Agency (ESA) mission Gaia,

processed by the Gaia Data Processing and Analysis Consortium (DPAC).

38

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